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A097075
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Expansion of (1-x-x^2)/(1-x-3*x^2-x^3).
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5
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1, 0, 2, 3, 9, 20, 50, 119, 289, 696, 1682, 4059, 9801, 23660, 57122, 137903, 332929, 803760, 1940450, 4684659, 11309769, 27304196, 65918162, 159140519, 384199201, 927538920, 2239277042, 5406093003, 13051463049, 31509019100, 76069501250
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OFFSET
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0,3
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COMMENTS
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Counts closed walks of length n at a vertex of a triangle, to which a loop has been added at one of the other vertices.
a(n) is the top left entry of the n-th power of the 3X3 matrix [0, 1, 1; 1, 1, 1; 1, 1, 0] or of the 3X3 matrix [0, 1, 1; 1, 0, 1; 1, 1, 1].
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LINKS
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Table of n, a(n) for n=0..30.
J. Bodeen, S. Butler, T. Kim, X. Sun, S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7
Index entries for linear recurrences with constant coefficients, signature (1,3,1).
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FORMULA
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a(n) = (1+sqrt(2))^n/4+(1-sqrt(2))^n/4+(-1)^n/2.
a(n) = a(n-1) + 3*a(n-2) + a(n-3).
a(n) = (-1)^n/2 + sum(k=0..floor(n/2), binomial(n, 2*k)*2^k)/2.
a(n) = (-1)^n/2 + A001333(n)/2.
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PROG
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(PARI) Vec((1-x-x^2)/(1-x-3*x^2-x^3) + O(x^50)) \\ Michel Marcus, Mar 25 2014
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CROSSREFS
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Cf. A000129, A051927, A097076.
Sequence in context: A121908 A231368 A245123 * A036673 A111189 A001004
Adjacent sequences: A097072 A097073 A097074 * A097076 A097077 A097078
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Jul 22 2004
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STATUS
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approved
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